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1

IIT-JEE 2002

Subjective
For any natural number $$m$$, evaluate
$$\int {\left( {{x^{3m}} + {x^{2m}} + {x^m}} \right){{\left( {2{x^{2m}} + 3{x^m} + 6} \right)}^{l/m}}dx,x > 0.} $$

Answer

$${1 \over 6}{{{{\left( {2{x^{3m}} + 3{x^{2m}} + 6{x^m}} \right)}^{{{m + 1} \over m}}}} \over {m + 1}} + C$$
2

IIT-JEE 2001

Subjective
Evaluate $$\int {{{\sin }^{ - 1}}\left( {{{2x + 2} \over {\sqrt {4{x^2} + 8x + 13} }}} \right)} \,dx.$$

Answer

$$\left( {x + 1} \right){\tan ^{ - 1}}\left( {{{2x + 2} \over 3}} \right) - {3 \over 4}\log \left( {4{x^2} + 8x + 13} \right) + C$$
3

IIT-JEE 1999

Subjective
Integrate $$\int {{{{x^3} + 3x + 2} \over {{{\left( {{x^2} + 1} \right)}^2}\left( {x + 1} \right)}}dx.} $$

Answer

$$ - {1 \over 2}\log \left| {x + 1} \right| + {1 \over 4}\log \left( {{x^2} + 1} \right) + {3 \over 2}{\tan ^{ - 1}}x + {x \over {1 + {x^2}}} + C$$
4

IIT-JEE 1996

Subjective
Evaluate $$\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx} $$.

Answer

$$\log \left( {{{1 + x{e^x}} \over {x{e^x}}}} \right) - {1 \over {1 + x{e^x}}} + C$$

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